The block Schur product is a Hadamard product


  • Erik Christensen



Given two $n \times n $ matrices $A = (a_{ij})$ and $B=(b_{ij}) $ with entries in $B(H)$ for some Hilbert space $H$, their block Schur product is the $n \times n$ matrix $ A\square B := (a_{ij}b_{ij})$. Given two continuous functions $f$ and $g$ on the torus with Fourier coefficients $(f_n)$ and $(g_n)$ their convolution product $f \star g$ has Fourier coefficients $(f_n g_n)$. Based on this, the Schur product on scalar matrices is also known as the Hadamard product.

We show that for a C*-algebra $\mathcal{A} $, and a discrete group $G$ with an action $\alpha _g$ of $G$ on $\mathcal{A} $ by *-automorphisms, the reduced crossed product C*-algebra $\mathrm {C}^*_r(\mathcal{A} , \alpha , G)$ possesses a natural generalization of the convolution product, which we suggest should be named the Hadamard product.

We show that this product has a natural Stinespring representation and we lift some known results on block Schur products to this setting, but we also show that the block Schur product is a special case of the Hadamard product in a crossed product algebra.


Bédos, E. and Conti, R., On discrete twisted $\rm C^*$-dynamical systems, Hilbert $\rm C^*$-modules and regularity, Münster J. Math. 5 (2012), 183–208.

Bédos, E. and Conti, R., Fourier series and twisted $\rm C^\ast $-crossed products, J. Fourier Anal. Appl. 21 (2015), no. 1, 32–75.

Christensen, E., Commutator inequalities via Schur products, in “Operator algebras and applications, the Abel Symposium 2015”, Abel Symp., vol. 12, Springer, 2017, pp. 133–149.

Christensen, E., On the complete boundedness of the Schur block product, Proc. Amer. Math. Soc. 147 (2019), no. 2, 523–532.

Christensen, E., Decompositions of Schur block products, J. Operator Theory (to appear).

Christensen, E. and Sinclair, A. M., Representations of completely bounded multilinear operators, J. Funct. Anal. 72 (1987), no. 1, 151–181.

Hadamard, J., Théorème sur les séries entières, Acta Math. 22 (1899), no. 1, 55–63.

Horn, R. A., The Hadamard product, in “Matrix theory and applications (Phoenix, AZ, 1989)”, Proc. Sympos. Appl. Math., vol. 40, Amer. Math. Soc., Providence, RI, 1990, pp. 87–169.

Horn, R. A. and Johnson, C. R., Topics in matrix analysis, Cambridge University Press, Cambridge, 1991.

Kadison, R. V. and Ringrose, J. R., Fundamentals of the theory of operator algebras. Vol. II: Advanced theory, Pure and Applied Mathematics, no. 100, Academic Press, Inc., Orlando, FL, 1986.

Livshits, L., Block-matrix generalizations of infinite-dimensional Schur products and Schur multipliers, Linear and Multilinear Algebra 38 (1994), no. 1-2, 59–78.

Mercer, R., Convergence of Fourier series in discrete crossed products of von Neumann algebras, Proc. Amer. Math. Soc. 94 (1985), no. 2, 254–258.

Pedersen, G. K., $C^*$-algebras and their automorphism groups, second ed., Pure and Applied Mathematics (Amsterdam), Academic Press, London, 2018.

Rørdam, M. and Sierakowski, A., Purely infinite $C^*$-algebras arising from crossed products, Ergodic Theory Dynam. Systems 32 (2012), no. 1, 273–293.

Schur, J., Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen, J. Reine Angew. Math. 140 (1911), 1–28.

Stinespring, W. F., Positive functions on $C^*$-algebras, Proc. Amer. Math. Soc. 6 (1955), 211–216.

Williams, D. P., Crossed products of $C^\ast $-algebras, Mathematical Surveys and Monographs, vol. 134, American Mathematical Society, Providence, RI, 2007.

Zeller-Meier, G., Produits croisés d'une $C^\ast $-algèbre par un groupe d'automorphismes, J. Math. Pures Appl. (9) 47 (1968), 101–239.



How to Cite

Christensen, E. (2020). The block Schur product is a Hadamard product. MATHEMATICA SCANDINAVICA, 126(3), 603–616.